Latent Poisson process and Laplace's rule

(Note: This is related to the post on Laplace’s rule).

For each interval length $s$ we have a sequence of binary variables $Y_1\left(s\right),Y_2\left(s\right),\dots Y_{n/s}\left(s\right)$. We want to connect these variables, for each $s$, in a meaningful way and apply Laplace’s rule. One way to do so is by assuming a latent Poisson process.

Let $X_t$ be a Poisson process with parameter $\lambda$, $t\in [0,n]$. Then the number of observations in the time frame $[t,t+s]$ is distributed as $$X_{t+s}-X_t\sim \text{Poisson}\left(\lambda s\right).$$ A natural way to make $X_{t+s}-X_t$ into a binary variable is to use $$Y_i\left(s\right)=1[X_{t+s}-X_t=0].$$ Then it follows that $$P\left(Y_i\right(s)=1)=e^{-\lambda s}.$$ To apply Laplace’s rule on a fixed time scale $s_0$, such as days, we’ll have to make the prior over $\pi \left(s_0\right)=e^{-\lambda s_0}$ uniform. This is equivalent to $\lambda s_0\sim \text{Exp}\left(1\right)$, or $\lambda \sim \text{Exp}\left(1/s_0\right)$.

Now, if we want to look at another time scale $s_1$, the induced prior for the success propability $\pi \left(s_1\right)=e^{-\lambda s_1}$ is $$\pi \left(s_1\right)=\pi \left(s_0\right){}^{\frac{s_1}{s_0}}\sim \text{Beta}(s_0/s_1,1).$$